As said in section , the form factor is underestimated due to the incorrect asymptotic behavior of the doubly off-shell form factor.
This was previously noted by [1]}, where it was pointed out that the correct asymptotic behavior can be recovered considering
two vector meson resonance multiplets instead of just one. Therefore, in order to give an estimation of the uncertainty stemming from this incorrect limit
we compute \(a_\mu ^{P,HLbL}\)  considering two multiplets and compare the result with that using one multiplet, both in the chiral limit. The values for
the fit parameters are kept identical for both estimations of \(a_\mu ^{P,HLbL}\) . For a more detailed description of the form factor and the short distance
constraints obtained among parameters we refer to [2]}. This uncertainty is estimated to be
\({\Delta a_\mu ^{P,HLbL}}_{\rm asym} = ^{+0.5}_{-0.0}\cdot 10^{-10}.\) 
